Question: William thought it would be nice to include $\frac{2}{21}$ of a pound of chocolate in each of the holiday gift bags he made for his friends and family. How many holiday gift bags could William make with $\frac{2}{3}$ of a pound of chocolate?
Solution: To find out how many gift bags William could create, divide the total chocolate ( $\frac{2}{3}$ of a pound) by the amount he wanted to include in each gift bag ( $\frac{2}{21}$ of a pound). $ \dfrac{{\dfrac{2}{3} \text{ pound of chocolate}}} {{\dfrac{2}{21} \text{ pound per bag}}} = {\text{ number of bags}} $ Dividing by a fraction is the same as multiplying by the reciprocal. The reciprocal of ${\dfrac{2}{21} \text{ pound per bag}}$ is ${\dfrac{21}{2} \text{ bags per pound}}$ $ {\dfrac{2}{3}\text{ pound}} \times {\dfrac{21}{2} \text{ bags per pound}} = {\text{ number of bags}} $ $ \dfrac{{2} \cdot {21}} {{3} \cdot {2}} = {\text{ number of bags}} $ Reduce terms with common factors by dividing the $2$ in the numerator and the $2$ in the denominator by $2$ $ \dfrac{{\cancel{2}^{1}} \cdot {21}} {{3} \cdot {\cancel{2}^{1}}} = {\text{ number of bags}} $ Reduce terms with common factors by dividing the $21$ in the numerator and the $3$ in the denominator by $3$ $ \dfrac{{1} \cdot {\cancel{21}^{7}}} {{\cancel{3}^{1}} \cdot {1}} = {\text{ number of bags}} $ Simplify: $ \dfrac{{1} \cdot {7}} {{1} \cdot {1}} = {7} $ William could create 7 gift bags.